$f\left(x\,y\,z\right)\=2\left(\sqrt{x^2\+z^2}\+x\+ y\right)$

and that the area function is

$A\left(x\,y\,z\right)\=2xy\+xz$ 

so that the constraint is $g\=A-50$.

Initialize

Loading Student:-MultivariateCalculus

$f\=2\left(\sqrt{x^2\+z^2}\+x\+ y\right)$$\stackrel{\text{assign}}{\to }$

$g\=2xy\+xz-50$$\stackrel{\text{assign}}{\to }$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Optimization≻Lagrange Multiplier Method

Table 4.9.4(a)   The Lagrange Multiplier Method task template

Implement the Lagrange multiplier method via first principles

$F\=f-\mathrm{\λ} g$$\stackrel{\text{assign}}{\to }$

$F$

$-\left(2xy\+xz-50\right)\mathrm{\λ}\+2\sqrt{x^2\+z^2}\+2x\+2y$

$\stackrel{\text{gradient}}{\to }$

$\stackrel{\text{to list}}{\to }$

$\left[-2xy-xz\+50\,-\left(2y\+z\right)\mathrm{\λ}\+\frac{2x}{\sqrt{x^2\+z^2}}\+2\,-2\mathrm{\λ}x\+2\,-x\mathrm{\λ}\+\frac{2z}{\sqrt{x^2\+z^2}}\right]$

$\stackrel{\text{solve}}{\to }$

$\left\{\mathrm{\λ}\=\frac{1}{10}-\frac{1}{10}\sqrt{3}\,x\=-5-5\sqrt{3}\,y\=-\frac{10}{3}\sqrt{3}\,z\=5\+\frac{5}{3}\sqrt{3}\right\}\,\left\{\mathrm{\λ}\=\frac{1}{10}\+\frac{1}{10}\sqrt{3}\,x\=5\sqrt{3}-5\,y\=\frac{10}{3}\sqrt{3}\,z\=5-\frac{5}{3}\sqrt{3}\right\}$

$\stackrel{\text{select entry 2}}{\to }$

$\left\{\mathrm{\λ}\=\frac{1}{10}\+\frac{1}{10}\sqrt{3}\,x\=5\sqrt{3}-5\,y\=\frac{10}{3}\sqrt{3}\,z\=5-\frac{5}{3}\sqrt{3}\right\}$

Optimization Assistant

The Optimization Assistant is no longer listed in Tools≻Assistants. It is now found in Tools≻Tutors≻Optimization

$\left(5\sqrt{3}-5\,\frac{10}{\sqrt{3}}\,5-\frac{5}{\sqrt{3}}\right)$ ≐ $\left(3.66\,5.77\,2.11\right)$

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$f≔2\left(\sqrt{x^2\+z^2}\+x\+ y\right)\:$

$g≔2xy\+xz-50\:$

Implement the Lagrange multiplier method

$\mathrm{LagrangeMultipliers}\left(f\,\left[g\right]\,\left[x\,y\,z\right]\right)$

$\left[\frac{21}{29}\,\frac{14}{29}\,-\frac{28}{29}\right]$

$$\begin{gathered} S≔\mathrm{simplify}\left(\left[\mathrm{LagrangeMultipliers}\left(f\,\left[g\right]\,\left[x\,y\,z\right]\,\mathrm{output}\=\mathrm{detailed}\right)\right]\right)\: \\ \mathrm{print}~\left(S\right)\left[\right] \end{gathered}$$

$\left[x\=-5-5\sqrt{3}\,y\=-\frac{10}{3}\sqrt{3}\,z\=5\+\frac{5}{3}\sqrt{3}\,{\mathrm{\λ}}_1\=\frac{1}{10}-\frac{1}{10}\sqrt{3}\,2\sqrt{x^2\+z^2}\+2x\+2y\=10-10\sqrt{3}\right]$

Implement the Lagrange multiplier method from first principles

$F≔f- \mathrm{\λ} g\:$

$$\begin{gathered} Q≔\mathrm{solve}\left(\mathrm{Equate}\left(\mathrm{Gradient}\left(F\,\left[x\,y\,z\,\mathrm{\λ}\right]\right)\,⟨0\,0\,0\,0⟩\right)\,\left[x\,y\,z\,\mathrm{\λ}\right]\,\mathrm{explicit}\right)\: \\ \mathrm{print}~\left(Q\right)\left[\right] \end{gathered}$$

$\left[x\=5\sqrt{3}-5\,y\=\frac{10}{3}\sqrt{3}\,z\=5-\frac{5}{3}\sqrt{3}\,\mathrm{\λ}\=\frac{1}{10}\+\frac{1}{10}\sqrt{3}\right]$